Transposed Poisson structures on solvable and perfect Lie algebras
Аннотация
Abstract We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e. on one-dimensional solvable extensions of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> -dimensional solvable extensions of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> -dimensional Heisenberg algebra; and on n -dimensional solvable extensions of the n -dimensional algebra with trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="fraktur">s</mml:mi> </mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and irreducible module gives a finite-dimensional Lie algebra with non-trivial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> -derivations, but without non-trivial transposed Poisson structures.